In this paper, we investigate the traveling wave solutions to a cubic predator-prey diffusion model with stage structure for the prey. Firstly, using the upper and lower solutions method we prove the existence and non-existence of weak traveling wave solutions. Furthermore, we prove that the weak traveling wave solutions are actually traveling wave solutions under additional conditions by using Lyapunov function method and LaSalle's invariance principle.

In this paper, we investigated the traveling wave solutions to a one prey-two competing predators model with nonlocal delay. First, we analyzed the stability of the positive equilibrium by using a Lyapunov function. Then, by examining the distribution of the roots of the characteristic equation, we ascertained the critical wave speed $c^{\ast }$. Finally, employing the cross iteration method and Schauder's fixed point theorem, we proved the existence of traveling wave solutions connecting the trivial equilibrium $(0,0,0)$ with the positive equilibrium $(u^{\ast },v^{\ast },w^{\ast })$ for wave speeds $c>c^{\ast }$. The incorporation of nonlocal delay into models featuring intra-specific and inter-specific competition significantly elevates computational complexity, thereby necessitating precise analytical estimates.